2) Power series Flashcards
What are the tests for absolute convergence of a series
What is the Cauchy-Hadamard test for convergence
What is the Leibniz test for alternating sequences
When does a sequence zn ∈ C converge
Let zn ∈ C and write zn = xn +i yn, xn, yn ∈ R. Then the sequence zn converges if and only if both xn and yn converge
What does it mean for a series ∑ zn in C to be absolutely convergent
What is the relationship between absolute convergence and convergence of a series in C
What happens when two absolutely convergent series in C are multiplied
What are the Ratio, Root, and Cauchy-Hadamard tests for convergence in C
What is a power series at z0 in C
What is the radius of convergence R of a power series, and what are its implications
What happens if a power series converges at w and ∣z∣<∣w∣
What is the radius of convergence and the disc of convergence of a power series
What are the formulas for finding the radius of convergence R of a power series
What is the relationship between the radii of convergence of a power series and its derivative
What can be said about the differentiability of a power series within its radius of convergence
What can be said about higher derivatives of a power series within its radius of convergence
What is the relationship between a power series and its integral within the radius of convergence
What can be concluded if two power series are equal within a disc of convergence
What determines the radius of convergence of the Taylor series of a function f(z) centered at z0
The radius of convergence of the Taylor series of f(z) centered at z0 is the distance from z0 to the nearest singularity of f(z)
How is the exponential function defined as a power series
What is the radius of convergence of the exponential function, and what does it imply
How are the complex sine and cosine functions defined as power series
What is Euler’s formula
How are the hyperbolic cosine and sine functions defined
What are the definitions of cosz, sinz, and their relationships to hyperbolic functions for z,w∈C
What is a period of a function
What is the definition of the complex logarithm
Let z ∈ C, z /= 0. Then a complex logarithm of z is log z = log|z| +i arg z where argz is any argument of z
What is the cut plane
What are the properties of the principal logarithm on the cut plane
The principal logarithm Log : C− → C is holomorphic
The derivative of the principal logarithm is 1/z
